Freiman ideals and the number of generators of powers of monomial ideals

Let lðIÞ denote the number of generators of a monomial ideal I. It is well
known that lðI
kÞ < lðI
kþ1Þ for k 0: In this paper we construct monomial
ideals I in F½x, y such that lðI
kþ1Þ < lðI
kÞ for all k l, given any positive
integer l. Also, we extend some results of Eliahou et al. by constructing
monomial ideals in R ¼ F½x1, :::, xn with lðI
2Þ < lðIÞ and investigate lðI
kÞ
for monomial ideals in R. Furthermore, we generalize the definition of
Freiman ideals given in Herzog and Zhu and extend some results with simpler proofs. In particular, we give a complete characterization of Freiman
ideals of maximum height in R